Averaging quarterly level data to the annual level

Answer to Q1: While imperfect, that approach is preferable out of the two you've presented here. Your 'implied' annual average will generally be higher than the actual annual average, with the extent of this error rising with employment churn. Consider calling the result the "quarterly earnings per employment, averaged for the year" or similar, so that your figure isn't misconstrued as annual earnings per employment. Or, if you have the time and inclination, you could report something more sophisticated than a ratio; consider an Autocorrelated Error Model, e.g. Chapter 16 of Griffiths et al, below.

Answer to Q2: This is often done when the data points are independent; the established techniques of “ratio estimation” are covered in, for example, Section 7.4 of John Rice’s textbook (below). However, your data aren't independent. You have four highly dependent readings of a two-dimensional signal (one dimension is earnings, one is employment). You’re quite right to be concerned about what you call double counting, for this particular problem.

Good luck!

References:

• The 3rd edition of John Rice’s Mathematical Statistics & Data Analysis (2007)
• Learning and Practicing Econometrics by W. Griffiths et al (1993)